Howdy, all. I am not sure if this is the right forum for this question. It isn't exactly an homework question, but it does stem very closely from a homework assignment in a first year graduate course in Abstract Algebra. The assignment has come and gone with limited success on my part, but some questions remain.(adsbygoogle = window.adsbygoogle || []).push({});

The problem was the well known one about a finite group G, |G| = pq where p<q primes. The object was to use Sylow Theorems to show that there is a unique (up to isomorphism) non-abelian group G if, and only if, p|q-1. Of course it is simple to show the case where p does not divide q-1, and my problem came from proving the other case. My question about that case is: Can one prove that there is such a unique non-abelian group without using any theory of automorphisms? Specifically is there a clever group action that can deliver the desired result?

The reason I ask is that our course has not mentioned automorphisms one bit. The text we are using is AshAbstract Algebra: The Basic Graduate Year, though most of our problems are pulled from Herstein'sTopics in Algebra. I have managed to read through the Herstein and pull out the tools I need for the problem, but could I have done the problem with the theory that has so far been provided? So far in a nutshell we have had the isomorphism theorems, the definition of group action, the orbit stabilizer theorem, and the class equation.

Thank you for your time. Cheers.

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# Finite non-abelian group of the order pq, pq primes

Can you offer guidance or do you also need help?

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